Trigonometry Examples

$\frac{\log_{6} (15 x)}{\log_{6} (5 x)} = 2$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$\log_{6} (5 x), 1$

Step 1.2

The LCM of one and any expression is the expression.

$\log_{6} (5 x)$

Step 2

Multiply each term in $\frac{\log_{6} (15 x)}{\log_{6} (5 x)} = 2$ by $\log_{6} (5 x)$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{\log_{6} (15 x)}{\log_{6} (5 x)} = 2$ by $\log_{6} (5 x)$ .

$\frac{\log_{6} (15 x)}{\log_{6} (5 x)} \log_{6} (5 x) = 2 \log_{6} (5 x)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $\log_{6} (5 x)$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{\log_{6} (15 x)}{\log_{6} (5 x)} \log_{6} (5 x) = 2 \log_{6} (5 x)$

Step 2.2.1.2

Rewrite the expression.

$\log_{6} (15 x) = 2 \log_{6} (5 x)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify $2 \log_{6} (5 x)$ by moving $2$ inside the logarithm.

$\log_{6} (15 x) = \log_{6} ({(5 x)}^{2})$

Step 2.3.2

Apply the product rule to $5 x$ .

$\log_{6} (15 x) = \log_{6} (5^{2} x^{2})$

Step 2.3.3

Raise $5$ to the power of $2$ .

$\log_{6} (15 x) = \log_{6} (25 x^{2})$

Step 3

Solve the equation.

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Step 3.1

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$15 x = 25 x^{2}$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Subtract $25 x^{2}$ from both sides of the equation.

$15 x - 25 x^{2} = 0$

Step 3.2.2

Factor the left side of the equation.

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Step 3.2.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$15 u - 25 u^{2} = 0$

Step 3.2.2.2

Factor $5 u$ out of $15 u - 25 u^{2}$ .

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Step 3.2.2.2.1

Factor $5 u$ out of $15 u$ .

$5 u (3) - 25 u^{2} = 0$

Step 3.2.2.2.2

Factor $5 u$ out of $- 25 u^{2}$ .

$5 u (3) + 5 u (- 5 u) = 0$

Step 3.2.2.2.3

Factor $5 u$ out of $5 u (3) + 5 u (- 5 u)$ .

$5 u (3 - 5 u) = 0$

Step 3.2.2.3

Replace all occurrences of $u$ with $x$ .

$5 x (3 - 5 x) = 0$

Step 3.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$3 - 5 x = 0$

Step 3.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.2.5

Set $3 - 5 x$ equal to $0$ and solve for $x$ .

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Step 3.2.5.1

Set $3 - 5 x$ equal to $0$ .

$3 - 5 x = 0$

Step 3.2.5.2

Solve $3 - 5 x = 0$ for $x$ .

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Step 3.2.5.2.1

Subtract $3$ from both sides of the equation.

$- 5 x = - 3$

Step 3.2.5.2.2

Divide each term in $- 5 x = - 3$ by $- 5$ and simplify.

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Step 3.2.5.2.2.1

Divide each term in $- 5 x = - 3$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 3}{- 5}$

Step 3.2.5.2.2.2

Simplify the left side.

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Step 3.2.5.2.2.2.1

Cancel the common factor of $- 5$ .

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Step 3.2.5.2.2.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 3}{- 5}$

Step 3.2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 5}$

Step 3.2.5.2.2.3

Simplify the right side.

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Step 3.2.5.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{3}{5}$

Step 3.2.6

The final solution is all the values that make $5 x (3 - 5 x) = 0$ true.

$x = 0, \frac{3}{5}$

Step 4

Exclude the solutions that do not make $\frac{\log_{6} (15 x)}{\log_{6} (5 x)} = 2$ true.

$x = \frac{3}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{5}$

Decimal Form:

$x = 0.6$